Nonlinear Schrodinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrodinger equations with combined power-type nonlinearities and harmonic potential.By constructing a variational problem the potential well method is applied.The structure of the potential well and the properties of the depth function are given.The invariance of some sets for the problem is shown.It is proven that,if the initial data are in the potential well or out of it,the solutions will lie in the potential well or lie out of it,respectively.By the convexity method,the sharp condition of the global well-posedness is given.